Probability

Notations

• $P(A|B)$, conditional probability of $A$ given the occurrence of $B$
• $A \cap B$, the intersection between A and B
• $A \cup B$, the union between A and B

Mutually Exclusive

Mutually exclusive identifies the relationships between multiple events, where these events cannot occur at in the same trial. Given two events ($A, B$) mutually exclusive to each other:

$$\begin{aligned} P(A \cap B) = 0 \end{aligned}$$

Examples

• Coin flip, occurrence of heads and tails are mutually exclusive.
• Number of boys or girls in a class, the number of boys in a finite total determine the number of girls in a class.

Independent

Independent identifies the relationships between multiple events, where the occurrence of one event does not affect the probability of the occurrence of other events. Given two events ($A, B$) independent to each other, the conditional probability satisfy the condition:

$$\begin{aligned} P(A | B) = \frac{P(A \cup B)}{P(B)} = P(A)\\ P(A)P(B) = P(A\cup B) \end{aligned}$$

Examples

• The probability of a asteroid colliding with Mars would not be affected by the occurrence of the national lottery, therefore these are independent events.